and
The first one is called the Comolet's diagram. It will
give us the final velocity of the bubble, thanks to its diameter and an
adimensionnal number called the Morton.
Now, let us explain the formulae that we need right there
to use this diagram.
First of all, V+ and d+ are not directly the speed and
the diameter of the bubble.
You may wonder what are their expressions:
and
You can also see that there is another number that we
have to take into account. It is called the Morton number. Physically speaking,
this number is constructed with the only properties of the fluid.
We can write it as Mo=[gravity x viscosity4]/[interfacial
tension3
x gravity2]
Another clearer expression for this Morton number is:
If you want the equations, here they are:
Stokes relation: 
if
d+ < 2Mo1/6
Wallis relation: 
if
2 Mo-1/8 <d+ <3,8 Mo1/14
Comolet relation: 
if
d+ >3,8 Mo1/14
Now, it is very simple to read the speed V+ as soon as
you have calculated the Morton's number and the diameter d+.
You choose your diameter on the horizontal axis, and
then, you follow the vertical axis until you meet the iso Morton curve
corresponding to the Morton's number you have previously calculated! Easy,
isn't it!
Let us demonstrate. We will do the calculation for the
bubble which diameter is 6 cm.
g=9,81m/s
rho#delta rho=1000kg/m3
Nu=10-6 m²/s
Sigma=0,074 N/m
D=6cm: And now, we find that d+ = 21,8 and Mo=2,42.10-11.
So, if you report those numbers on the previous graphic,
you will get a V+ equal to something around 3 or 4.
You can get a more precise value for V+ with the formulae
given: We see that d+>0,66=d+ >3,8 Mo1/14 so we use the Comolet
relation. The result is V+ = 3,34.
Then you invert the formula to get Vinfinite
thanks to V+ and you find that: Vinfinite#3,6m/s.
You can also say, when regarding the diagram, that the shape of the bubble is probably something between an ellipse and a parabola.
D=1cm: d+=3,63and this is once again the comolet relation that we have to use.
It gives us V+=1,56, so Vinfinite=1,68m/s.
The second one gives us the same kind of results, but you have to calculate another number called the Bond number. It represents the relative importance of Gravity compared to the one of interfacial tension.
Here is the expression for the Bond number: 
Now here is the diagram:
So, if we take back our two previous examples, we find that the Bond number is equal to:
D=6cm: Bo=486
D=1cm: Bo=13,51
We can notice that the bigger the bubble is, the more important the gravity forces is. As a matter of fact, Archimed's force grows up with the size of the bubble, thus explaining this phenomena.
So, now that we have calculated the Bond number we can
easily find out the right position on the diagram and then determine the
Reynolds number that corresponds to our case. The velocity is then known
through the Reynolds number!
Here, we will just look at the shape of the bubble, considering
the fact that its velocity has been determined previously with the Comolet's
diagram.
You can see that the bubble of 1 cm diameter has a strange form, like previously seen, between an ellipse and a parabola, whereas the bubble of 6 cm diameter is clearly a parabola or an inverted saucer.
The last thing that seems to be interesting, when dealing with adimensionnal numbers, is to define the Weber number, which is defined this way:
Just to mention it, we calculate the Weber for our 1cm diameter bubble.
The Weber number is equal to We=380.
* First of all, we have been surprised
to obtain so good results with Fluent concerning the final velocity of
the bubble. As a matter of fact, for the bubble of 1cm diameter, Fluent
gives us a velocity of 1,2 to 1,3 m/s, whereas the velocity we were expecting
a velocity around 1,7 m/s according to the theory.
So, you can notice that the difference
is around 20%, which is quite good!
Nonetheless, our domain is not
very long, and the velocity of the bubble is very likely to still evolve
after the end of our calculation. Yet, what we can say is that it seems
that at the end of our calculation time, the shape of our bubble appears
to be very close to the final one. So it is possible that the error between
the model calculated with Fluent and the one predicted by the theory is
greater than the one given upper.
* Concerning the small bubble (D= 5 mm), it remains circular, what is predicted by the theory. As a matter of fact, when you look at the diagrams, with such a bubble you are in the zone of spherical bubble : that is what we notice in the presentation of the results.
* As for the biggest one, the error we have made about the boundary conditions have been fatal to our calculations, and considering the fact that one calculation requires a dozen hours, we haven't made another attempt.
* At last the square bubble is a
funny case, but we have to admit that it has no physical existence and
no other interest than the one of curiosity. But, this put in evidence
the fact that interfacial tension is well modelized in Fluent, because
the bubble tends to become spherical what is its equilibrium shape, considering
the beginning of the simulation.
